• Korean Journal of Mathematics
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• v.16 no.3
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• pp.369-378
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• 2008

The aim of this paper is to study the stability problem for the pexider type trigonometric functional equation f(x + y) − f(x−y) = 2g(x)h(y), which is related to the d'Alembert, the Wilson, the sine, and the mixed trigonometric functional equations.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.235-243
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• 2009

In this paper we will investigate the superstability of the generalized d'Alembert type functional equations ${\sum}^m_{i=1}f(x+{{\sigma}^i}(y))$ = kg(x)f(y) and ${\sum}^m_{i=1}f(x+{{\sigma}^i}(y))$ = kf(x)g(y).

• Korean Journal of Mathematics
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• v.16 no.1
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• pp.103-114
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• 2008

In this paper, we investigate the superstability problem for the pexiderized cosine functional equations f(x+y) +f(x−y) = 2g(x)h(y), f(x + y) + g(x − y) = 2f(x)g(y), f(x + y) + g(x − y) = 2g(x)f(y). Consequently, we have generalized the results of stability for the cosine($d^{\prime}Alembert$) and the Wilson functional equations by J. Baker, $P.\;G{\check{a}}vruta$, R. Badora and R. Ger, and G.H. Kim.

• The Pure and Applied Mathematics
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• v.17 no.4
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• pp.397-411
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• 2010

In this paper, we will investigate the superstability for the sine functional equation from the following Pexider type functional equation: $f(x+y)-g(x-y)={\lambda}{\cdot}h(x)k(y)$ ${\lambda}$: constant, which can be considered an exponential type functional equation, the mixed functional equation of the trigonometric function, the mixed functional equation of the hyperbolic function, and the Jensen type equation.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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• pp.1-18
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• 2012

The aim of this paper is to investigate the superstability of the pexider type sine(hyperbolic sine) functional equation $f(\frac{x+y}{2})^{2}-f(\frac{x+{\sigma}y}{2})^{2}={\lambda}g(x)h(y),\;{\lambda}:\;constant$ which is bounded by the unknown functions ${\varphi}(x)$ or ${\varphi}(y)$. As a consequence, we have generalized the stability results for the sine functional equation by P. M. Cholewa, R. Badora, R. Ger, and G. H. Kim.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.185-195
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• 2008

We consider a system of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions and Gelfand generalized functions. As a consequence we find locally integrable solutions of the n-dimensional trigonometric functional equation.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.711-722
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• 2020

Given a semigroup S generated by its squares equipped with an involutive automorphism 𝝈 and a multiplicative function 𝜇 : S → ℂ such that 𝜇(x𝜎(x)) = 1 for all x ∈ S, we determine the complex-valued solutions of the following functional equations f(xy) + 𝜇(y)f(𝜎(y)x) = 2f(x)g(y), x, y ∈ S and f(xy) + 𝜇(y)f(𝜎(y)x) = 2f(y)g(x), x, y ∈ S.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1157-1169
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• 2016

We find the distributional solutions of the Wilson's functional equations $$u{\circ}T+u{\circ}T^{\sigma}-2u{\otimes}v=0,\\u{\circ}T+u{\circ}T^{\sigma}-2v{\otimes}u=0,$$ where $u,v{\in}{\mathcal{D}}^{\prime}({\mathbb{R}}^n)$, the space of Schwartz distributions, T(x, y) = x + y, $T^{\sigma}(x,y)=x+{\sigma}y$, $x,y{\in}{\mathbb{R}}^n$, ${\sigma}$ an involution, and ${\circ}$, ${\otimes}$ are pullback and tensor product of distributions, respectively. As a consequence, we solve the $Erd{\ddot{o}}s$' problem for the Wilson's functional equations in the class of locally integrable functions. We also consider the Ulam-Hyers stability of the classical Wilson's functional equations $$f(x+y)+f(x+{\sigma}y)=2f(x)g(y),\\f(x+y)+f(x+{\sigma}y)=2g(x)f(y)$$ in the class of Lebesgue measurable functions.

• Journal for History of Mathematics
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• v.34 no.5
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• pp.153-164
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• 2021

A functional equation is an equation which is satisfied by a function. Some elementary functional equations can be manipulated with elementary algebraic operations and functional composition only. However to solve such functional equations, somewhat critical and creative thinking ability is required, so that it is educationally worth while teaching functional equations. In this paper, we look at the origin of functional equations, and their characteristics and educational meaning and effects. We carefully suggest the use of the functional equations as a material for school mathematics education.